AP Calculus 7.1 Worksheet All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. Fill in the blanks.

a) Integrating velocity gives _____________________

b) Integrating the absolute value of velocity gives _________________________.

c) New position = ____________________ + _________________________. d) In general, integrating a rate (say of clowns per car) gives you _________________________________. 2. A particle moves along the x-axis (units in cm). Its initial position at t = 0 sec is x (0) = 15. The figure shows the graph of the particle’s velocity v (t). The numbers are the areas of the enclosed regions. a) Describe the motion of the particle (include the position and when the particle moving right, left, and stopped) b) What is the particle’s displacement between t = 0 and t = c ? c) What is the total distance traveled by the particle in the same time period? d) Give the positions of the particle at times a, b, and c. e) Approximately where does the particle achieve its greatest positive acceleration on the interval [0, b] ?

3. [Calculator] Pollution is being removed from a lake at a rate modeled by the function 0.520 ty e-= tons/yr, where t is the number

of years since 1995. How much pollution was removed from the lake between 1995 and 2005.

a b c 5

7

14

v (t)

4. [Calculator] The rate at which customers arrive at a counter to be served is modeled by the function F defined by

( ) 12 6 cost

F tp

æ ö÷ç= + ÷ç ÷çè ø

for 0 < t < 60, where F (t) is measured in customers per minute and t is measured in minutes. a) To the nearest whole number, how many customers arrive at the counter over the 60-minute period? b) What is the average number of customers served each minute over the 60-minute period? 5. [Calculator] A particle moves along the x – axis so that its velocity at time t is given by

( ) ( )2

1 sin2

tv t t

æ ö÷ç ÷=- + ç ÷ç ÷çè ø.

At time t = 0, the particle is at position x = 1.

a) Find the acceleration of the particle at time t = 2. Is the speed of the particle increasing at t = 2? Why or why not?

b) Find all times t in the open interval 0 < t < 3 when the particle changes direction. Justify your answer.

c) Find the total distance traveled by the particle from time t = 0 until time t = 3.

d) During the time interval 0 < t < 3, what is the greatest distance between the particle and the origin? Show the work that leads to your answer.

6. [Calculator] The rate of consumption of oil in the United States during the 1980s (in billions of barrels per year) is modeled by the

function /2527.08 tC e= , where t is the number of years after January 1, 1980. Find the total consumption of oil in the United States from January 1, 1980 to January 1, 1990.

7. Tickets to a concert were sold out in 24 hours. The graph below shows the rate at which people bought tickets to the concert during the 24-hour period. Before the tickets went on sale, 100 tickets had been set aside for VIPs. a) How does the graph indicate the total number of tickets purchased in the 24 hours. b) Approximate the total tickets purchased in the last 24 hours. c) If all ticket holders show up to the concert, how many people will attend? 8. [Calculator] The rate at which people enter an amusement park on a given day is modeled by the function E defined by

( ) 2

15600

24 160E t

t t=

- +.

The rate at which people leave the same amusement park on the same day is modeled by function L defined by

( ) 2

9890

38 370L t

t t=

- +.

Both E (t) and L (t) are measured in people per hour and time t is measured in hours after midnight. These functions are valid for 9 < t < 23, the hours during which the park is open. At time t = 9, there are no people in the park.

a) How many people have entered the park by 5:00 pm (t = 17) ? Round your answer to the nearest whole number.

b) The price of admission to the park is $15 until 5:00 pm (t = 17) . After 5:00 pm, the price of admission to the park is $11. How many dollars are collected from admissions to the park on the given day? Round your answer to the nearest dollar.

c) Let ( ) ( ) ( )( )9

t

H t E x L x dx= -ò for 9 < t < 23. The value of ( )17H to the nearest whole number is 3725.

Find the value of ( )' 17H and explain the meaning of ( )17H and ( )' 17H in the context of the park.

d) At what time t, for 9 < t < 23, does the model predict that the number of people in the park is a maximum?

10. [Calculator] Population density measures the number of people per square mile inhabiting a given living area. The population of a local town decreases as you move away from the city center can be approximated by the function 10,000(3 – r) at a distance r miles from the city center. a) If the population density approaches zero at the edge of the city, what is the city’s radius? b) A thin ring around the center of the city has thickness r and radius r. (Hint: Let r = distance from center of city to “middle of ring”) Draw a picture of this and determine the area of the thin ring.

c) Explain why the population of the ring in part b is approximately ( )( )10000 3 2r r rp- .

d) Estimate the total population of the town by setting up and evaluating a definite integral. 11. Each graph shown below shows the velocity (in m/sec) of a particle moving on the x-axis. Each particle starts at x = 2 when t = 0. a) Find where each particle is at the end of the trip. b) Find the total distance traveled by each particle. 12. [True or False] If the velocity of a particle moving along the x-axis is always positive, then the displacement and the total distance traveled are equal.

AP Calculus 7.2 Worksheet All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. Find the area of the shaded region analytically (without a calculator). a) b) c)

2. [No Calculator] Find the area of the region bounded by the graphs of ( ) 21 2f x x x= + - and ( ) 1g x x= - .

y = 0.5sec2 (x)

y = –4sin2 (x)

x = y2

x = y3

x = 12y2 – 12y3

x = 2y2 – 2y

3. [No Calculator] Find the area of the region bounded by the graphs of ( ) 4

2g x

x=

-, y = 4, and x = 0.

4. [No Calculator] Consider the area of the region bounded by the graphs of 3y x= and y = x .

a) Explain why the area cannot be found by the single integral ( )1

3

1

x x dx-

-ò .

b) Write an expression involving a single integral that DOES represent the area of the region?

5. [Calculator] Find the area of the region between the graphs of ( ) 3 23 10f x x x x= - - and ( ) 2 2g x x x=- + .

6. [No Calculator] Find the area of the shaded region analytically.

y = 1

y = x2/4

y = x

7. [No Calculator] The region bounded below by the parabola y = x2 and above by the line y = 4 is to be partitioned into two subsections of equal area by cutting across it with the horizontal line y = c. a) Sketch the region and draw a line y = c across it that looks about right. In terms of c, what are the coordinates of the points where the line and the parabola intersect? b) Find c by integrating with respect to y. (This puts c in the limits of integration.)

8. [Calculator] Let R be the shaded region enclosed by the graphs of 2xy e-= , ( )sin 3y x=- , and the y-axis as

shown in the figure below. Find the area of R.

A Little AP Preparation …

T (hours)

0 2 5 7 8

E (t) (hundreds of entries)

0 4 13 21 23

[Calculator] A zoo sponsored a one-day contest to name a new baby elephant. Zoo visitors deposited entries in a special box between noon t 0and 8 P.M. t 8. The number of entries in the box t hours

after noon is modeled by a differentiable function E for 0 t 8. Values of Et , in hundreds of entries, at various times t are shown in the table above.

(a) Use the data in the table to approximate the rate, in hundreds of entries per hour, at which entries were being deposited at time t 6. Show the computations that lead to your answer.

(b) Use a trapezoidal sum with the four subintervals given by the table to approximate the value

of ( )8

0

1

8E t dtò . Using correct units, explain the meaning of ( )

8

0

1

8E t dtò in terms of the number of entries.

(c) At 8 P.M., volunteers began to process the entries. They processed the entries at a rate modeled by the function P, where ( ) 3 230 298 976P t t t t= - + - hundreds of entries per hour for 8 t 12.

According to the model, how many entries had not yet been processed by midnight t 12? (d) According to the model from part (c), at what time were the entries being processed most

quickly? Justify your answer.

AP Calculus 7.3 Worksheet (Day 1) All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. [No Calculator] The base of a solid is the region enclosed by the graph of xy e-= , the coordinate axes, and the line x = 3. If all

plane cross sections perpendicular to the x – axis are squares, then its volume is

A) 61

2

e--

B) 61

2e-

C) 6e-

D) 3e-

E) 31 e--

2. [Calculator] The base of a solid S is the region enclosed by the graph of lny x= , the line x = e, and the x-axis. If the cross

sections of S perpendicular to the x – axis are semicircles, then the volume of S is 3. The base of a solid is the region bounded by the lines ( ) 21 xf x = - , ( ) 21 xg x =- + , and x = 0. If the cross sections perpendicular

to the x-axis are equilateral triangles, find the volume of the solid.

4. [Calculator] Let R be the region enclosed by the graphs of ( )2ln 1y x= + and y = cos x.

a) Find the area of R. b) The base of a solid is the region R. Each cross section of the solid perpendicular to the x-axis is an equilateral triangle. Write an expression involving one or more integrals that gives the volume of the solid. Do not evaluate. 5. The solid lies between planes perpendicular to the x-axis at x = 0 and x = 4. The cross sections perpendicular to the axis on the

interval 0 < x < 4 are squares whose diagonals run from y x= to y x=- . Find the volume of the solid.

6. The solid lies between planes perpendicular to the y-axis at y = 0 and y = 2. The cross sections perpendicular to the y-axis are

circular (NOT semicircular) disks with diameters running from the y-axis to the parabola 25x y= . Find the volume of the solid.

7. The base of the solid is the disk 2 2 1x y+ £ . The cross sections are perpendicular to the y-axis between y = 1 and y = –1 are

isosceles right triangles with one leg in the disk. Find the volume of the solid.

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AP Calculus 7.3 Worksheet (Day 2) All work must be shown in this course for full credit. Unsupported answers may receive NO credit.

1. Suppose region R is in the first quadrant bounded by the graphs of 3y x= and x = 8.

a) If R is rotated about the x-axis find the resulting volume. b) Find the volume if R is rotated around the line x = 8. 2. The region in the first quadrant bounded by the graph of y = sec x, 4x p= , and the axes is rotated about the x-axis. What is the

volume of the solid generated?

A) 2

4p

B) 1p-

C) p

D) 2p

E) 83p

3. The volume of the solid obtained by revolving the region enclosed by the ellipse 2 29 9x y+ = about the x-axis is

A) 2p

B) 4p

C) 6p

D) 9p

E) 12p

4. Le

the h

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A little AP Preparation … [Calculator] The rate at which people enter an auditorium for a rock concert is modeled by the function R given by

( ) 2 31380 675R t t t= - for 0 < t < 2 hours; R t is measured in people per hour. No one is in the auditorium at time t 0,

when the doors open. The doors close and the concert begins at time t 2. a) How many people are in the auditorium when the concert begins? b) Find the time when the rate at which people enter the auditorium is a maximum. Justify your answer. c) The total wait time for all the people in the auditorium is found by adding the time each person waits, starting at the time the person enters the auditorium and ending when the concert begins. The function w models the total wait time for all the people who enter the auditorium before time t. The derivative of w is given by ( ) ( )' (2 )w t t R t= - .

Find w (2) – w (1), the total wait time for those who enter the auditorium after time t 1. d) On average, how long does a person wait in the auditorium for the concert to begin? Consider all people who enter the auditorium after the doors open, and use the model for total wait time from part c.

AP Calculus 7.3 Worksheet (day 3) All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. Let R be the region between the graphs of y = 1 and y = sin x from x = 0 to 2x p= . What is the volume of the solid obtained by

revolving R about the x-axis? 2. The region enclosed by the graph of 2y x= , the line x = 2, and the x-axis is revolved about the y-axis. What is the volume of the

solid generated? 3. [Calculator] A region in the first quadrant is enclosed by the graphs of 2xy e= , x = 1, and the coordinate axes. If the region is

rotated about the y-axis, what is the volume of the solid generated?

4. [Calculator] Let R be the region in the first quadrant enclosed by the graph of ( )1

31y x= + , the line x = 7, the x-axis, and the y-

axis. What is the volume of the solid generated when R is revolved about the y-axis?

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AP Calculus 7.3 Worksheet (day 4) All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. Let R be the region bounded by the graphs of 2 4y x= + , y = 8, and x = 0, set up and evaluate the integral that gives the volume of the solid generated by revolving R about the y – axis. a) Use the disc method b) Use the shell method The following two questions were done on the last worksheet using 2 rectangles and the washer method. Try doing them again using the shell method. Compare your answers. 2. [Calculator] A region in the first quadrant is enclosed by the graphs of 2xy e= , x = 1, and the coordinate axes. If the region is rotated about the y-axis, what is the volume of the solid generated?

3. [Calculator] Let R be the region in the first quadrant enclosed by the graph of ( )1

31y x= + , the line x = 7, the x-axis,

and the y-axis. What is the volume of the solid generated when R is revolved about the y-axis?

4. Find the volume of the solid formed by revolving the region bounded by the graphs of y = x and 24y x x= - about the y-axis. 5. Set up an integral and use your calculator to find the volume of the solid generated by revolving the region bounded by y x= and the lines y = 2 and x = 0 about … (I would draw a different picture each time)

a) … the x-axis. b) … the y-axis. c) … the line y = 2 d) … the line x = 4. 6. The shaded region R, shown in the figure below, is rotated about the y-axis to form a solid who volume is 10 cubic inches. Of the following, which best approximates k? A) 1.51 B) 2.09 C) 2.49 D) 4.18 E) 4.77

x

y

k

7. [Calculator] Let R be the region in the first and second quadrants bounded above by the graph of 2

201

yx

=+

and

below by the horizontal line y = 2. a) Find the area of R. b) Find the volume of the solid generated when R is rotated about the x-axis. c) Find the volume of the solid generated when R is rotated about the line y = 25. d) Find the volume of the solid generated when R is rotated about the line x = –4. e) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are semicircles. Find the volume of this solid.

A little AP Preparation … [Calculator] The amount of water in a storage tank, in gallons, is modeled by a continuous function on the time interval 0 < t < 7, where t is measured in hours. In this model, rates are given as follows: i) The rate at which water enters the tank is

( ) ( )2100 sinf t t t= gallons per hour for 0 < t < 7.

ii) The rate at which water leaves the tank is

( )250 for 0 32000 for 3 7

tg t

t

ìïïíïïî

£ <=

< £ gallons per hour.

The graphs of f and g, which intersect at t = 1.617 and t = 5.076, are shown in the figure to the right. At time t = 0, the amount of water in the tank is 5000 gallons. a) How many gallons of water enter the tank during the time interval 0 < t < 7 ? Round your answer to the nearest gallon. b) For 0 < t < 7, find the time intervals during which the amount of water in the tank is decreasing. Give a reason for your answer. c) For 0 < t < 7, at what time t is the amount of water in the tank the greatest? To the nearest gallon, compute the amount of water at this time. Justify your answer.